Without expanding, show that the values of each of the following determinants are zero:(i) 82712351643(ii) 6-322-12-1052(iii) 23713175152012(iv) 1/aa2bc1/bb2ac1/cc2ab(v) a+b2a+b3a+b2a+b3a+b4a+b4a+b5a+b6a+b(vi) 1aa2-bc1bb2-ac1cc2-ab(vii) 491639742623(viii) 0xy-x0z-y-z0(ix) 143673543172(x) 12223242223242523242526242526272(xi) abca+2xb+2yc+2zxyz(xii) 2x+2-x22x-2-x213x+3-x23x-3-x214x+4-x24x-4-x21(xiii) sinαcosαcos(α+δ)sinβcosβcos(β+δ)sinγcosγcos(γ+δ)(xiv) sin223°sin267°cos180°-sin267°-sin223°cos2180°cos180°sin223°sin267°(xv) cosx+y-sinx+ycos2ysinxcosxsiny-cosxsinx-cosy(xvi) 23+35515+465103+115155(xvii) sin2AcotA1sin2BcotB1sin2CcotC1, where A, B, C are the angles of ∆ABC.

Without expanding, show that the values of each of the following deter

1.	Without expanding, show that the values of each of the following determinants are zero:(i) 82712351643(ii) 6-322-12-1052(iii) 23713175152012(iv) 1/aa2bc1/bb2ac1/cc2ab(v) a+b2a+b3a+b2a+b3a+b4a+b4a+b5a+b6a+b(vi) 1aa2-bc1bb2-ac1cc2-ab(vii) 491639742623(viii) 0xy-x0z-y-z0(ix) 143673543172(x) 12223242223242523242526242526272(xi) abca+2xb+2yc+2zxyz(xii) 2x+2-x22x-2-x213x+3-x23x-3-x214x+4-x24x-4-x21(xiii) sinαcosαcos(α+δ)sinβcosβcos(β+δ)sinγcosγcos(γ+δ)(xiv) sin223°sin267°cos180°-sin267°-sin223°cos2180°cos180°sin223°sin267°(xv) cosx+y-sinx+ycos2ysinxcosxsiny-cosxsinx-cosy(xvi) 23+35515+465103+115155(xvii) sin2AcotA1sin2BcotB1sin2CcotC1, where A, B, C are the angles of ∆ABC.
Answer» Without expanding, show that the values of each of the following determinants are zero: (i) $\|\begin{matrix} 8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3 \end{matrix}\|$ (ii) $\|\begin{matrix} 6 & - 3 & 2 \\ 2 & - 1 & 2 \\ - 10 & 5 & 2 \end{matrix}\|$ (iii) $\|\begin{matrix} 2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12 \end{matrix}\|$ (iv) $\|\begin{matrix} 1 / a & a^{2} & b c \\ 1 / b & b^{2} & a c \\ 1 / c & c^{2} & a b \end{matrix}\|$ (v) $\|\begin{matrix} a + b & 2 a + b & 3 a + b \\ 2 a + b & 3 a + b & 4 a + b \\ 4 a + b & 5 a + b & 6 a + b \end{matrix}\|$ (vi) $\|\begin{matrix} 1 & a & a^{2} - b c \\ 1 & b & b^{2} - a c \\ 1 & c & c^{2} - a b \end{matrix}\|$ (vii) $\|\begin{matrix} 49 & 1 & 6 \\ 39 & 7 & 4 \\ 26 & 2 & 3 \end{matrix}\|$ (viii) $\|\begin{matrix} 0 & x & y \\ - x & 0 & z \\ - y & - z & 0 \end{matrix}\|$ (ix) $\|\begin{matrix} 1 & 43 & 6 \\ 7 & 35 & 4 \\ 3 & 17 & 2 \end{matrix}\|$ (x) $\|\begin{matrix} 1^{2} & 2^{2} & 3^{2} & 4^{2} \\ 2^{2} & 3^{2} & 4^{2} & 5^{2} \\ 3^{2} & 4^{2} & 5^{2} & 6^{2} \\ 4^{2} & 5^{2} & 6^{2} & 7^{2} \end{matrix}\|$ (xi) $\|\begin{matrix} a & b & c \\ a + 2 x & b + 2 y & c + 2 z \\ x & y & z \end{matrix}\|$ (xii) $\|\begin{matrix} {(2^{x} + 2^{- x})}^{2} & {(2^{x} - 2^{- x})}^{2} & 1 \\ {(3^{x} + 3^{- x})}^{2} & {(3^{x} - 3^{- x})}^{2} & 1 \\ {(4^{x} + 4^{- x})}^{2} & {(4^{x} - 4^{- x})}^{2} & 1 \end{matrix}\|$ (xiii) $\|\begin{matrix} \sin α & \cos α & \cos (α + δ) \\ \sin β & \cos β & \cos (β + δ) \\ \sin γ & \cos γ & \cos (γ + δ) \end{matrix}\|$ (xiv) $\|\begin{matrix} \sin^{2} 23 ° & \sin^{2} 67 ° & \cos 180 ° \\ - \sin^{2} 67 ° & - \sin^{2} 23 ° & \cos^{2} 180 ° \\ \cos 180 ° & \sin^{2} 23 ° & \sin^{2} 67 ° \end{matrix}\|$ (xv) $\|\begin{matrix} \cos (x + y) & - \sin (x + y) & \cos 2 y \\ \sin x & \cos x & \sin y \\ - \cos x & \sin x & - \cos y \end{matrix}\|$ (xvi) $\|\begin{matrix} \sqrt{23} + \sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{15} + \sqrt{46} & 5 & \sqrt{10} \\ 3 + \sqrt{115} & \sqrt{15} & 5 \end{matrix}\|$ (xvii) $\|\begin{matrix} \sin^{2} A & \cot A & 1 \\ \sin^{2} B & \cot B & 1 \\ \sin^{2} C & \cot C & 1 \end{matrix}\|, where A, B, C are the angles of ∆ A B C .$

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